Christian Schoof

Ice-sheet acceleration driven by melt supply variability

Increased ice velocities in Greenland are contributing significantly to eustatic sea level rise. Faster ice flow has been associated with ice–ocean interactions in water-terminating outlet glaciers and with increased surface meltwater supply to the ice-sheet bed inland. Observed correlations between surface melt and ice acceleration have raised the possibility of a positive feedback in which surface melting and accelerated dynamic thinning reinforce one another, suggesting that overall warming could lead to accelerated mass loss. Here I show that it is not simply mean surface melt but an increase in water input variability that drives faster ice flow. Glacier sliding responds to melt indirectly through changes in basal water pressure, with observations showing that water under glaciers drains through channels at low pressure or through interconnected cavities at high pressure. Using a model that captures the dynamic switching between channel and cavity drainage modes, I show that channelization and glacier deceleration rather than acceleration occur above a critical rate of water flow. Higher rates of steady water supply can therefore suppress rather than enhance dynamic thinning, indicating that the melt/dynamic thinning feedback is not universally operational. Short-term increases in water input are, however, accommodated by the drainage system through temporary spikes in water pressure. It is these spikes that lead to ice acceleration, which is therefore driven by strong diurnal melt cycles and an increase in rain and surface lake drainage events rather than an increase in mean melt supply.

The effect of cavitation on glacier sliding

Basal sliding is one of the most important components in the dynamics of fast–flowing glaciers, but remains poorly understood on a theoretical level. In this paper, the problem of glacier sliding with cavitation over hard beds is addressed in detail. First, a bound on drag generated by the bed is derived for arbitrary bed geometries. This bound shows that the commonly used sliding law, τb = CumbNn, cannot apply to beds with bounded slopes. In order to resolve the issue of a realistic sliding law, we consider the classical Nye–Kamb sliding problem, extended to cover the case of cavitation but neglecting regelation. Based on an analogy with contact problems in elasticity, we develop a method which allows solutions to be constructed for any finite number of cavities per bed period. The method is then used to find sliding laws for irregular hard beds, and to test previously developed theories for calculating the drag generated by beds on which obstacles of many different sizes are present. It is found that the maximum drag attained is controlled by those bed obstacles which have the steepest slopes.

Marine ice-sheet dynamics. Part 1. The case of rapid sliding

Marine ice sheets are continental ice masses resting on bedrock below sea level. Their dynamics are similar to those of land-based ice sheets except that they must couple with the surrounding floating ice shelves at the grounding line, where the ice reaches a critical flotation thickness. In order to predict the evolution of the grounding line as a free boundary, two boundary conditions are required for the diffusion equation describing the evolution of the grounded-ice thickness. By analogy with Stefan problems, one of these conditions imposes a prescribed ice thickness at the grounding line and arises from the fact that the ice becomes afloat. The other condition must be determined by coupling the ice sheet to the surrounding ice shelves. Here we employ matched asymptotic expansions to study the transition from ice-sheet to ice-shelf flow for the case of rapidly sliding ice sheets. Our principal results are that the ice flux at the grounding line in a two-dimensional ice sheet is an increasing function of the depth of the sea floor there, and that ice thicknesses at the grounding line must be small compared with ice thicknesses inland. These results indicate that marine ice sheets have a discrete set of steady surface profiles (if they have any at all) and that the stability of these steady profiles depends on the slope of the sea floor at the grounding line.

On the mechanics of ice-stream shear margins

We investigate the mechanics of ice-stream shear margins based on the assumption that the underlying bed behaves plastically. Sliding is assumed to occur if a prescribed, locally defined yield stress is attained, while no sliding is assumed possible if basal shear stress is lower than the yield stress. Mathematically, the ice-flow problem takes the form of a contact problem, in which the zones of sliding are part of the solution and cannot be prescribed arbitrarily. Simplistic assumptions about the location of till failure, or about mechanical conditions at the bed, predict stress singularities at the margins which lead to corresponding singularities in the basal melt rate. The ice-flow problem is solved using a complex variable method, and an associated quasi-static thermal problem is also solved using a Greens function. High stress concentrations, which coincide with high rates of strain heating, are found on the ice-stream side of the margins, where basal melting is also greatest. Our results further indicate that a temperate zone may form over time above the bed in the margins. These findings differ from earlier studies based on dif- ferent sliding laws, suggesting a high sensitivity of margin behaviour to basal conditions.

Neural Networks Applied to Estimating Subglacial Topography and Glacier Volume

Abstract To predict the rate and consequences of shrinkage of the earth’s mountain glaciers and ice caps, it is necessary to have improved regional-scale models of mountain glaciation and better knowledge of the subglacial topography upon which these models must operate. The problem of estimating glacier ice thickness is addressed by developing an artificial neural network (ANN) approach that uses calculations performed on a digital elevation model (DEM) and on a mask of the present-day ice cover. Because suitable data from real glaciers are lacking, the ANN is trained by substituting the known topography of ice-denuded regions adjacent to the ice-covered regions of interest, and this known topography is hidden by imagining it to be ice-covered. For this training it is assumed that the topography is flooded to various levels by horizontal lake-like glaciers. The validity of this assumption and the estimation skill of the trained ANN is tested by predicting ice thickness for four 50 km × 50 km regions that...

Basal perturbations under ice streams: form drag and surface expression

Classical sliding theories consider ice sliding over obstacles which are much shorter than the thickness of overlying ice. Here we present a theory which considers form drag generated under ice streams by large obstacles such as subglacial bedforms, which may have lengths comparable to ice thickness. We also investigate how perturbations at the surface of an ice stream can be generated by such bedforms, and develop a mathematical framework for separating the effects of such local (kilometre-scale) variations in ice flow from the bulk flow of the ice stream.

Variational methods for glacier flow over plastic till

We investigate the mechanics of ice streams and glaciers flowing over a bed consisting of Coulomb-plastic subglacial sediment, or more generally, of channel flows with Coulomb or ‘solid’ friction laws at the boundary. Sliding is assumed to occur if shear stress at the glacier bed attains a prescribed, locally defined yield stress, while no sliding is assumed possible below that yield stress. Importantly, the location of regions of slip and no slip at the bed is not known a priori , but forms part of the solution. By analogy with friction problems in elasticity, we derive a weak formulation as a semi-coercive variational inequality, which admits a unique solution provided a solvability condition ensuring force balance is satisfied. The variational formulation is then exploited to calculate numerical solutions, and we investigate the effect of variations in subglacial water pressure, ice thickness and surface slope on the discharge of a valley glacier with a plastic bed. Significant differences are found between the behaviour of wide and narrow as well as steep and shallow-angled glaciers, and our results further indicate the need to develop models capable of accounting for longitudinal stresses.

Stick-slip motion of an Antarctic Ice Stream: The effects of viscoelasticity

Stick-slip behavior is a distinguishing characteristic of the flow of Whillans Ice Stream (Siple Coast, Antarctica). Distinct from stick slip on Northern Hemisphere glaciers, which is generally attributed to supraglacial melt, the behavior is thought be controlled by basal processes and by tidally induced stress. However, the connection between stick-slip behavior and flow of the ice stream on long time scales, if any, is not clear. To address this question we develop a new ice flow model capable of reproducing stick-slip cycles similar to ones observed on the Whillans Ice Plain. The model treats ice as a viscoelastic material and emulates the weakening and healing that are suggested to take place at the ice-till interface. The model results suggest the long-term ice stream flow that controls ice discharge to surrounding oceans is somewhat insensitive to certain aspects of stick-slip behavior, such as velocity magnitude during the slip phase and factors that regulate it (e.g., elastic modulus). Furthermore, it is found that factors controlling purely viscous flow, such as temperature, influence stick-slip contribution to long-term flow in much the same way. Additionally, we show that viscous ice deformation, traditionally disregarded in analysis of stick-slip behavior, has a strong effect on the timing of slip events and therefore should not be ignored in efforts to deduce bed properties from stick-slip observations.

On the granular stress-geometry equation

Using discrete calculus, we derive the missing stress-geometry equation for rigid granular materials in two dimensions, in the mean-field approximation. We show that i) the equation imposes that the voids cannot carry stress, ii) stress transmission is generically elliptic and has a quantitative relation to anisotropic elasticity, and iii) the packing fabric plays an essential role.

The role of subtemperate slip in thermally-driven ice stream marginmigration

The amount of ice discharged by an ice stream depends on its width, and the widths of unconfined ice streams such as the Siple Coast ice streams in West Antarctica have been observed to evolve on decadal to centennial timescales. Thermally-driven widening of ice streams provides a mechanism for this observed variability through melting of the frozen beds of adjacent ice ridges. This widening is driven by the heat dissipation in the ice stream margin, where strain rates are high, and at the bed of the ice ridge, where subtemperate sliding is possible. The inflow of cold ice from the neighboring ice 5 ridges impedes ice stream widening. Determining the migration rate of the margin requires resolving conductive and advective heat transfer processes on very small scales in the ice stream margin, and these processes cannot be resolved by large scale ice sheet models. Here, we exploit the thermal boundary layer structure in the ice stream margin to investigate how the migration rate depends on these different processes. We derive a parameterization of the migration rate in terms of parameters that can be estimated from observations or large scale model outputs, including the lateral shear stress in the ice stream margin, the ice 10 thickness of the stream, the influx of ice from the ridge, and the bed temperature of the ice ridge. This parameterization will allow the incorporation of ice stream margin migration into large-scale ice sheet models.